This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

Taking a Die for a Walk

Stage: 1 and 2 Challenge Level:

Taking a Die for a Walk

We often roll a die to find a number from $1$ to $6$ for a game that we're playing. We wait until the die stops and then see what number is on top. What about if we slow it down and have the die slowing rolling along? What if we make it even slower and we move the die to a new face one at a time?

In this way we can move the die forward/backward or right/left. You might like to think about it as North, South, East or West.

You can then move the die around and take it on a journey. In this investigation, we're going to think of a journey as never re-visiting a place where the die has already been. So it can't cross back over its path or go back over where it has just been.

When I first thought about this I thought of the die leaving a kind of "footprint" of the bottom number wherever it went on its journey. You can do this if you want to.

I then found it easier to look at the number on the top as each new "step" was taken.

So with a die which might show:-

- or maybe you have one with numbers on it:-

we can go on a little journey:-

So, starting with $1$ on the top we move down. (I make sure that $2$ is going to come up by moving the die around before starting.) I then got $2$ followed by $6$. I then rolled it over to the right a few times. $3$ shows up, then $1$ again, followed by $4$.

Another journey could be:-

We start off as before. But, when I've rolled to the right a couple of times, I go up and it shows a $2$, and then left and it shows a $3$. It's a bit difficult to see where I've gone so I've added an arrow to show the direction.

What I CANNOT do now it to go further by rolling left from that last $3$ which would give a $5$ on top of the $2$ I had from my first step. Remember - no crossing back again!

So, explore a little and see what things happen as you roll around different paths and you write the numbers down.

What patterns can you see in the arrangements of numbers?

Perhaps you can begin to predict what number will appear by visualising the die rolling before you check with a real die.

Why do this problem?

This problem could be used when the four directions, north, south, east and west are being introduced or discussed. It could also be helpful if learners are apprehensive about number work as it gives an easy "open door" to number exploration and enquiry. Its openess means that it can be approached by a very wide
attainment range. This problem presents an interesting context for encouraging children to visualise the die and predict the path of numbers.

Possible approach

You could start with the group sitting around in a circle and using a big die to roll in the middle. Invite learners to tell you what they know about dice, perhaps by asking questions such as, "If $6$ is on top, what number is on the bottom?". What do the children notice about opposite faces of the die? (They add to $7$.)

When you get to introducing the actual activity there is a need to move the die carefully so that the footprint is seen to be in a grid formation. If you are involving the four directions, north, south, east and west, it would be a good idea to have a sheet with the compass points on it. Stress the rule that the die may not cross back over its path or go back over where it has just been. It
would help to have a large sheet of paper also on the floor where you can record the path of the die. As you roll the die, you could ask children to talk to a partner and predict the number which is about to appear on the top. This visualisation can be quite tricky and it requires a good understanding of the features of dice.

Set the class off on exploring their own paths, giving them each a die and squared paper. It would be an advantage if they worked in pairs so that they are able to talk through their ideas with a partner. Depending on how they get on, it might be worth stopping them after some time to share what they have been doing so far. At this stage, you could choose to focus in on a particular idea for
everyone to explore which has been stimulated by one or more pairs. Warn the group that you will expect them to feed back to everyone at the end of the session.

Once they have had more time to work, bring the whole group together once more to show their various "dice walks" and to discuss what they have discovered. What do they notice about the patterns they have created? They could compare, for example, adjacent numbers in two different "walks". What is the same and what is different about them? Can they give explanations for their observations?

Key questions

What numbers have you found?

Is the die going north, south, east or west now?

You seem to have a system for doing this, can you tell me about it?

What do you notice about the number patterns?

Can you explain why that happens?

Possible extension

These sheets give many ideas about how the investigation could be taken further. You may want to give them out to children, or simply suggest something else for them to pursue. Also go Inky Cubes

Possible support

Some pupils might find it helpful to start with the die rolling in just one direction to get the idea of how to record and to become more familiar with the way a die is constructed. Larger dice and paper with $2$ cm squares might help those with difficulties in using fine motor skills.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.